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arXiv · 2111.09766

Untangling Circular Drawings: Algorithms and Complexity

Sujoy Bhore·Guangping Li·Martin Nöllenburg·Ignaz Rutter·Hsiang-Yun Wu
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Abstract

We consider the problem of untangling a given (non-planar) straight-line circular drawing $\delta_G$ of an outerplanar graph $G=(V, E)$ into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph $G$, it is clear that such a crossing-free circular drawing always exists and we define the circular shifting number shift$(\delta_G)$ as the minimum number of vertices that are required to be shifted in order to resolve all crossings of $\delta_G$. We show that the problem Circular Untangling, asking whether shift$(\delta_G) \le K$ for a given integer $K$, is NP-complete. For $n$-vertex outerplanar graphs, we obtain a tight upper bound of shift$(\delta_G) \le n - \lfloor\sqrt{n-2}\rfloor -2$. Based on these results we study Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all the crossings. In this case, we provide a tight upper bound shift$(\delta_G) \le \lfloor \frac{n}{2} \rfloor-1$ and present a constructive polynomial-time algorithm to compute the circular shifting number of almost-planar drawings.

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Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, Hsiang-Yun Wu. 2021-11-18. Untangling Circular Drawings: Algorithms and Complexity. https://doi.org/10.4230/lipics.isaac.2021.19

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